A different model of random simplicial complexes was studied by M. Kahle [1]}, [2]} and by some other authors.
These are clique complexes of random Erdös - Rényi graphs; here one takes a random graph in the
Erdös - Rényi model and declares as a simplex every subset of vertices which form a clique, i.e. such that every two vertices of the subset are connected by an edge.
Compared with the Linial - Meshulam model, the clique complex has “randomness”  in dimension one but it influences the structure in all the higher dimensions.
